Fermat's Last Theorem

This is the one case Fermat himself wrote a proof for, found among his papers after his death. The method — infinite descent — builds, from any hypothetical solution, a strictly smaller one, impossible for positive integers; it is the ancestor of well-founded induction used throughout modern mathematics and computer science.

Suppose a solution in positive integers to $x^4+y^4=z^2$ exists; choose one with $z$ minimal. If $d=\gcd(x,y)>1$, then $d^2\mid z$ and $(x/d)^4+(y/d)^4=(z/d^2)^2$ is smaller — contradiction. So $\gcd(x,y)=1$, $(x^2,y^2,z)$ is a primitive Pythagorean triple; say $x$ is odd. By Euclid's parametrization, coprime $m>n>0$ of opposite parity give $x^2=m^2-n^2$, $y^2=2mn$, $z=m^2+n^2$.

From $x^2+n^2=m^2$, the triple $(x,n,m)$ is itself primitive, so coprime $a>b>0$ give $x=a^2-b^2$, $n=2ab$, $m=a^2+b^2$. Then $y^2=4mab$, so $(y/2)^2=mab$ with $m,a,b$ pairwise coprime and product a perfect square, forcing each to be a square: $m=z_1^2$, $a=x_1^2$, $b=y_1^2$.

Substituting into $m=a^2+b^2$ gives $z_1^2=x_1^4+y_1^4$ — a new solution of the same equation, with $z_1\le z_1^2=m<m^2+n^2=z$: strictly smaller, contradicting minimality. No solution exists. For $x^4+y^4=z^4$, setting $Z=z^2$ would give a solution of $x^4+y^4=Z^2$, just ruled out.

A simple algebraic trick reduces the infinitely many exponents to two families — exponent 4 and odd prime exponents — but even so, three centuries of the sharpest elementary techniques conquered only a handful of primes at a time; the theorem was finally settled only by importing entirely new machinery from elliptic curves.

Every integer $n>2$ is divisible by an odd prime $p\ge3$ (write $n=pk$) or is a power of 2 with $n\ge4$, hence divisible by 4 ($n=4k$). A solution of $x^n+y^n=z^n$ would give, with $X=x^k,Y=y^k,Z=z^k$, a solution of $X^p+Y^p=Z^p$ or $X^4+Y^4=Z^4$. So FLT for all $n>2$ follows from the cases $n=4$ (proved above) and every odd prime $p\ge3$.

Euler (1770) extended Fermat's descent to prove $p=3$ elementarily. Over the following century, Germain, Legendre and Kummer proved larger and larger classes of primes — Kummer's 1850s work settled every "regular" prime — but no single descent handled every prime, and large primes resisted every attempt for another 140 years.

The case $p \ge 5$ was settled in 1994-95 by Andrew Wiles with Richard Taylor, using ideas outside elementary number theory. Given a hypothetical $a^p+b^p=c^p$, Gerhard Frey (1984) associated the elliptic curve $y^2 = x(x-a^p)(x+b^p)$; Kenneth Ribet (1990) proved this curve, if it existed, could not be modular. Wiles then proved every semistable elliptic curve over the rationals is modular, so the Frey curve cannot exist, and no such $a,b,c$ exist. This full argument — Galois representations, deformation rings, modular forms — is reconstructed step by step in the companion proof "Wiles's modularity proof via the Taylor-Wiles method (1994)" filed under the great problem Fermat's Last Theorem in this library; it needs machinery introduced only later in the curriculum, so it is not repeated here.